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On the application of the SCD semismooth* Newton method to solving Stokes problem with stick-slip boundary conditions

V. Arzt, P. Beremlijski, H. Gfrerer, J. V. Outrata

Abstract

The paper deals with the 3D Stokes problem with Navier-Tresca stick-slip boundary conditions. A weak formulation of this problem leads to a variational inequality of the second kind, coupled with an equality constraint. This problem is then approximated using the mixed finite element method, yielding a generalized equation, to the numerical solution of which we implement a variant of the SCD semismooth* Newton method. This includes also a globalization technique ensuring convergence for arbitrary starting points. Numerical experiments demonstrate the effeciency of this approach.

On the application of the SCD semismooth* Newton method to solving Stokes problem with stick-slip boundary conditions

Abstract

The paper deals with the 3D Stokes problem with Navier-Tresca stick-slip boundary conditions. A weak formulation of this problem leads to a variational inequality of the second kind, coupled with an equality constraint. This problem is then approximated using the mixed finite element method, yielding a generalized equation, to the numerical solution of which we implement a variant of the SCD semismooth* Newton method. This includes also a globalization technique ensuring convergence for arbitrary starting points. Numerical experiments demonstrate the effeciency of this approach.

Paper Structure

This paper contains 8 sections, 5 theorems, 77 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Stokes problem
  3. Continuous and two-field weak formulation of the problem
  4. Mixed finite element method
  5. Algebraic formulation
  6. SCD semismooth Newton* method
  7. On the SCD semismooth$^*$ Newton method for the Stokes problem with Navier-Tresca boundary condition
  8. Numerical experiments

Key Result

Theorem 1

Let $f\in(L^2(\Omega))^3,\ \sigma_N\in\left(L^2(\gamma_N)\right)^3$, and $g,\kappa\in L^\infty(\gamma_S)$, $g\geq0$, $\kappa\geq 0$. Then the solution $(u,p)$ to eq1:2 exists, and the velocity component $u$ is unique. If $\gamma_N\neq \emptyset$, then the pressure component $p$ is unique as well, an

Figures (6)

  • Figure 1: Scheme of $\Omega$
  • Figure 2: Cube mesh
  • Figure 3: $\hbox{g}=0$
  • Figure 4: $\hbox{g}=5$
  • Figure 5: 2D scheme of flow in Cerebral Aneurysm
  • ...and 1 more figures

Theorems & Definitions (13)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 2
  • Definition 4: GfrOut22
  • remark 1
  • Theorem 3
  • Lemma 4
  • proof
  • ...and 3 more